Question

A firm in an oligopolistic industry has the following demand and total cost equations:

P = 600 – 20Q  and  TC = 700 + 160Q + 15Q²

c. price and quantity that maximizes revenue at which profit will be at least $580. Fully and completely explain why the price and quantity you choose maximize revenue given the constraint of a minimum profit.

Answer 1

P = 600 - 20Q

Profit must greater than equal to $ 580

In case of profit maximization MR must be equal to MC

TR = PQ =(600 -20Q)Q = 600Q - 20Q^2

MR = 600 - 40Q

MC = 160 + 30Q

MR = MC

600 -40Q = 160 +30Q

600 - 160 = 40Q + 30Q

70Q = 440

Q = 440/70 = 6.28 units

P = 600 - 20×6.28 = 600 - 125.6 = $474.4/unit

Minimum profit of $580 Thus this equation must hold true

TR -TC =580

600Q-20Q^2 - 700 -160Q - 15Q^2 = 580

440Q - 35Q^2 -700 = 580

35Q^2 -440Q + 1280 = 0

7Q^2 - 88Q + 256 = 0

7Q^2 - 56Q - 32Q + 256 =0

7Q(Q-8)-32(Q-8) = 0

(7Q -32)(Q-8) = 0

EITHER, Q = 8 OR, Q = 32/7 = 4.57

Profit MAXIMIZING output is 6.28 units therefore Q = 4.57 must be rejected.

In order to earn a minimum profit of $580

Quantity must be 8 units

Price = 600 - 20×8 = 600 - 160 = $ 440/ unit